It is shown that there is a
unique group G with property I : G is a central extension of Z2 by the Hall-Janko
group of order 604,800 in which a 7-Sylow subgroup S7 is normalized by an element
of order four. Also, G has a six-dimensional complex representation X. The
proof is rather round-about. First, it is shown that there are at most two
six-dimensional linear groups X(G) projectively representing the Hall-Janko
group, and all such linear groups are algebraically conjugate. The character
table and generators found by M. Hall for G∕Z(G) are used. It is shown
that a linear group L over GF(9) coming from the one candidate for a six
dimensional group projectively representing the Hall-Janko group actually satisfies
property I. This is done by showing that L has a permutation representation on
100 three-dimensional subspaces of (GF(9))6 and the image is permutation
isomorphic to Hall’s permutation group. Hall later studied the geometry of these
subspaces. In the course of constructing the character table of any group G
with property I,G was found to have a sixdimensional representation. Once
this representation is known to exist, it is possible to give two easier ways
of constructing generators. The faithful characters on G are given in the
appendix with only one representative of each pair of algebraically conjugate
characters.
